Bayes Linear Methods I
Adjusting Beliefs: Concepts and Properties
Michael Goldstein
University of Durham
[B/D] Home Page: http://maths.dur.ac.uk/stats/bd/
Abstract
Bayes linear methodology provides a quantitative structure for expressing our beliefs and systematic methods for
revising these beliefs given observational data. Particular emphasis is placed upon interpretation of and diagnostics for
the specification. The approach is similar in spirit to the standard Bayes analysis, but is constructed so as to avoid much
of the burden of specification and computation of the full Bayes case. This report is the first of a series describing Bayes
linear methods. In this document, we introduce some of the basic machinery of the theory. Examples, computational
issues, detailed derivations of results and approaches to belief elicitation will be addressed in related reports.
1
Introduction
Bayes linear methodology provides a quantitative structure for expressing our beliefs and systematic methods for revising
these beliefs given observational data. Particular emphasis is placed upon interpretive and diagnostic features of the
analysis. The approach is similar in spirit to the standard Bayes analysis, but is constructed so as to avoid much of the
burden of specification and computation of the full Bayes case. From a foundational view, the Bayes analysis emerges as
a special case of the Bayes linear approach. From a practical view, Bayes linear methods offer a way of tackling problems
which are too complex to be handled by standard Bayesian tools.
This report is the first of a series describing Bayes linear methods. In this document, we introduce some of the basic
machinery of the theory. Examples, computational issues, detailed derivations of results and methods for belief elicitation
will be addressed in related reports. In particular, [9] contains a simple tutorial guide to the material in this report, by
means of a simple example, with details as to how the relevant calculations may be programmed in the computer language
[B/D].
We cover the following material.
Section 2 concerns our basic approach to quantifying uncertainty and details the specification requirements for the Bayes
linear analysis.
Section 3 defines and interprets the notions of adjusted expectation and adjusted variance for a collection of quantities,
and explains the role of canonical directions in summarising the effects of an adjustment.
Section 4 concerns the types of diagnostic comparisons that we may make after we have evaluated the belief adjustment.
In particular, we discuss the role of the bearing of the adjustment in summarising the overall magnitude and nature
of the changes between prior and adjusted beliefs.
Section 5 covers the role of partial adjustments for analysing beliefs which are modified in stages.
Section 6 Bayes linear methods are so named as, formally, they derive their properties from the linear structure of inner
product spaces rather than the boolean structure of probability spaces. This section summarises the geometry
underlying the adjustment of beliefs.
1
2
Quantifying uncertainty
In a quantitative belief analysis, we quantify various aspects of our beliefs about a collection of unknown quantities, and
then, typically, we use further information to modify our statements of belief about these quantities. In this section, we
consider the structure within which we shall express initial uncertainties.
2.1
Quantifying uncertainty
There are many different ways in which beliefs may be quantified. Most familiar, perhaps, is the Bayesian approach, in
which beliefs about all of the uncertain quantities of interest are represented in terms of a joint probability distribution.
In practice, the specification of such a joint probability distribution will often be largely arbitrary due to the difficulty
that most of us find in thinking meaningfully and consistently in high numbers of dimensions (or even in low numbers
of dimensions - indeed even specifying a single probability may be a daunting task if our answer really matters for some
purpose).
Full probabilistic specification is unwieldy as a fundamental expression of prior knowledge in that it requires such an
extremely large number of statements of prior knowledge, expressing judgements to so fine a level of detail, that usually
we have neither the interest nor the ability to make most of these judgements in a meaningful way. To escape from the
straitjacket of full probabilistic specification, we suggest an approach which is related in spirit to the Bayesian approach,
but is more straightforward to apply.
Suppose, therefore, that we intend to quantify some aspects of our prior judgements. It is reasonable to require that
our subsequent analysis should only be based on those aspects of our beliefs which we are both willing and able to specify.
Each number that we specify expresses some aspect of our prior knowledge, and as such requires careful consideration. Our
concern is to develop a methodology which allows us to specify and analyse relatively small, carefully chosen collections
of quantitative judgements about whichever aspects of a problem are within our ability to specify in a meaningful way.
We begin by describing our basic approach to the quantification of belief.
2.2
Expectation
When we reduce the number of aspects of uncertainty about which specifications are to be made, we may also simplify the
nature of the specification process, by using methods which lead directly to the particular quantifications that we require.
For this purpose, we make direct assessments for our (subjective) expectations for the various uncertainties of interest.
The idea of treating expectation as a primitive quantity and specifying expectation directly rather than through some
intermediary probabilistic specification has been developed at length by various authors. The most detailed exposition of
this approach is described in de Finetti ([1, 2]). De Finetti uses the term prevision for an expectation elicited directly and
suggests various operational definitions for directly elicited expectations.1 In this formulation, the probability of an event
is simply the expectation or prevision for the associated indicator function.
We shall therefore assume, in what follows, that we have made various prior expectation statements, through direct
elicitation. We cannot give formal rules for specifying prior expectations any more than we can give such rules for
specifying prior probabilities in a standard Bayes analysis. Each expectation expresses a subjective choice that must be
made given our assessment of the situation in question. Our account concerns the various methods by which we can
improve our quantifications of belief, given such initial judgements and relevant data. Thus, while the forming of sensible
prior judgements is of fundamental importance, it falls outside the strict remit of this account. We will discuss in a separate
report the issues involved in eliciting such restricted prior specifications. All that we shall observe here is that, because
any full probability specification over some outcome space is logically equivalent to a specification of the expectation for
every random quantity which could possibly be constructed over that outcome space, it must be a substantially easier task
to make a careful prior specification of the expectations only for a small subset of such quantities.
2.3
Belief specification
In general, the level of detail at which we choose to describe our beliefs will depend on
• how interested we are in the various aspects of the problem;
1 For example, the simplest such definition is that your prevision for a random quantity X is the value x that you consider to be a “fair price” for a
ticket which pays X .
2
• our ability to specify each aspect of our uncertainty;
• the amount of time and effort that we are willing to expend on the problem;
• how much detail is required from our prior specification in order to extract the necessary information from the data.
We must, therefore, recognise that our analysis depends not only upon the observed data but also upon the level of
detail to which we have expressed our beliefs. The formal framework within which we shall express our judgements is as
follows.
We begin by supplying an ordered (finite or infinite) list C = {X 1 , X 2 , . . .} of random quantities, for which we shall
make statements of uncertainty. We call C the base for our analysis.
For each X i , X j ∈ C we specify
1. the expectation, E(X i ), giving a simple quantification of our belief as to the magnitude of X i ;
2. the variance, Var(X i ), quantifying our uncertainty or degree of confidence in our judgements of the magnitude of
Xi ;
3. the covariance, Cov(X i , X j ), expressing a judgement on the relationship between the quantities, quantifying the
extent to which observation on X j may (linearly) influence our belief as to the size of X i .
These expectations, variances and covariances are specified directly, although this does not preclude us from deducing
the values from some larger specification, or even, when this is practical, from a full prior probability distribution. We
require that each element of C must have finite prior variance.
For any ordered subcollections, A, B, of elements of C, we write
Var(A)
to denote the variance matrix of the vector of elements of A, and we write
Cov(A, B)
to denote the covariance matrix between the vectors A and B.
We control the level of detail of our investigations by our choice of the collection C. The most detailed collection that
we could possibly select would consist of the indicator functions for all of the combinations of possible values of all of
the quantities of interest.With this choice of C, we obtain a full probability specification over some underlying outcome
space. Sometimes this special case may be appropriate, but for large problems we will usually restrict attention to small
subcollections of this collection. (Thus, for example, if there were two quantities Y and Z which we might measure, then C
might contain the terms {Y, Y 2 , Z , Z 2 , Y Z }.) It is preferable to work explicitly with the collection of belief specifications
that we have actually made rather than to pretend to specify much larger collections of prior belief statements.
2.4
Belief structures
The formal structure which is described by our belief specification is as follows. We have a collection of random quantities
C = {X 1 , X 2 , . . .}, each with finite prior variance. We construct the linear space hCi consisting of all finite linear
combinations
h 0 X 0 + h 1 X i1 + . . . + h k X ik
of the elements of C, where X 0 is the unit constant. We view hCi as a vector space in which each X i is a vector, and
linear combinations of vectors are the corresponding linear combinations of the random quantities. hCi is in general the
largest structure over which expectations are defined once we have defined expectations for the elements of C.
Covariance defines an inner product (·, ·) and norm over hCi, defined, for X, Y ∈ hCi to be
(X, Y ) = Cov(X, Y ), kX k2 = Var(X ).
3
The vector space, hCi, with the covariance inner product (., .), defines an inner product space, which we denote [C].
We call [C] a belief structure with base {C}.2 In this space, the ‘length’ of any vector is equal to the standard deviation
of the random quantity.
A belief structure provides the minimal formal structuring for a belief specification which is sufficient for our general
analyses. A traditional discrete probability space is represented within this formulation by a base consisting of indicator
functions over a partition, so that the vectors are the linear combinations of the indicator functions, or, equivalently, the
random variables over the probability space. A continuous probability specification is expressed as the Hilbert space of
square integrable functions over the space with respect to the prior measure. In the probability specification, all covariances
between all such pairs of random quantities over the space must be specified. The belief structure allows us to restrict, by
our choice of base, the specification to any linear subspace of this collection, so that we may specify only those aspects of
our beliefs which we are both able and willing to quantify. Therefore, the formal properties of our approach follow from
the linearity underlying the inner product structure, which is why we term our approach Bayes linear.
In the following sections, we describe various general properties of belief adjustment. In the final section, we return
to the geometry underlying this approach, and describe the formal structure of the analysis.
3
Adjusting beliefs by data
In this section, we discuss the adjustment of a collection of expectation statements, given data. As this report is intended
as a summary of concepts and properties, results will be stated without proof. Technical details will be discussed in a
separate report. To simplify the exposition, we shall suppose that our chosen base C contains a finite number of quantities.
In the final section, we will describe the underlying geometry, identify the equivalent results for infinite collections, and
give geometric explanations for the various properties that we have described.
3.1
Adjusted expectation
We have a collection, C, of random quantities, for which we have specified prior means, variances and covariances.
Suppose now that we observe the values of a subset, D = {D1 , . . . , Dk }, of the members of C. We intend to modify our
beliefs about the remaining quantities, B = {B1 , . . . , Br }, in C. A simple method by which we can modify our prior
expectation statements is to evaluate the adjusted expectation for each quantity.
The adjusted expectation of a random quantity X ∈ B, given observation of a collection of quantities D, written
E D (X ), is defined to be the linear combination
E D (X ) = h TD D =
k
X
h i Di
i=0
which minimises
E((X −
k
X
h i Di )2 ),
i=0
over all collections h = (h 0 , h 1 , . . . , h k ), where D0 = 1. E D (X )is sometimes called the Bayes linear rule for X
given D.
E D (X ) is determined by the prior mean, variance and covariance specifications. If Var(D) is of full rank3 then
E D (X ) = E(X ) + Cov(X, D)[Var(D)]−1 (D − E(D)).
(1)
Adjusted expectation obeys the following properties:
1. for any quantities X 1 , X 2 and constants c, d we have,
E D (cX 1 + d X 2 ) = cE D (X 1 ) + dE D (X 2 )
(2)
2 Strictly, the inner product space is defined over the closure of the equivalence classes of random quantities which differ by a constant, so that we
identify any vector, such as X 0 , with zero variance with the zero vector.
3 If Var(D) is not of full rank, then we may discard elements of D so that the reduced collection is of full rank. Otherwise, we may consider
[Var(D)]−1 to be the Moore-Penrose generalised inverse in the following matrix equations.
4
2. for any X , we have
E(E D (X )) = E(X )
3.2
(3)
Interpretation
How should we interpret adjusted expectations? There are four inter-related interpretations that we can offer.
• The simplest interpretation is to view the quantity E D (X ) as an ‘estimator’ of the value of X, which combines the data
with simple aspects of our prior beliefs in an intuitively plausible manner and which leads to a useful methodology.
Alternately, if we have extensive data sources to draw upon, then we may construct our prior judgements from these
sources and use our approach to develop ‘estimators’ which can be viewed as complementary to certain standard
estimators in multivariate analysis.
• The second interpretation is to view adjusted expectation simply as a primitive which quantifies certain aspects of
our beliefs, in a similar manner to the original expectation statement. Indeed, in de Finetti’s formal development
of prevision, the principle operational definition that he offers is that our prevision for X is the value x which we
would choose if we were forced to suffer a penalty
L = k(X − x)2 ,
(4)
where k is a constant defining the units of loss. In this view adjusted expectation simply expresses the extension of
our choice of preferences from the certain choice x to the random choice
L D = k(X −
k
X
xi Di )2 .
(5)
i=0
In the special case where the elements Di are the indicator functions for a partition, then this is equivalent to de Finetti’s
choice for the operational definition of the conditional prevision, xi , of X given each event Di . Under this view,
adjusted expectations are simply informative summaries, generalising the corresponding conditional expectations
defined over indicator functions.
• If we are committed in principle to a full Bayes view based on complete probabilistic specification of all uncertainties,
then we may view adjusted expectations as offering simple tractable approximations to a full Bayes analysis for
complicated problems. In addition, the various interpretive measures and diagnostic tests which we shall develop
below offer insights which are relevant to any full Bayes analysis.
• We have described three alternative views of adjusted expectation, each of which has merit in certain contexts and
reflects the contrasting views that may be held concerning the revision of beliefs. However, we hold a fourth view,
which, by proceeding directly by foundational arguments, subsumes each of the above views. This view explains
why we should view adjusted expectation as a primitive, the precise sense in which adjusted expectation may be
viewed as an ‘estimator’, and the general properties which may be claimed for the estimate. Further, it reverses our
third interpretation above by identifying a full Bayes analysis as a simple special case of the general analysis which
we advocate.
Our immediate intention is to describe the practical machinery of our approach. Therefore, we do not at this point intend
to take logical and philosophical diversions into foundational issues, and we shall develop the formal relationship between
belief adjustment and belief revision elsewhere. Instead, for now we will move between the first three interpretations that
we have listed above, viewing adjusted expectation as an intuitively plausible numerical summary statement about our
beliefs given the data. There is no implication that this value will fully express our genuine revised belief concerning the
expectation of X . Rather, we have been explicit as to precisely which aspects of our prior beliefs have been utilised in
order to assess the adjusted expectation. As with any other formal analysis that we might carry out, adjusted expectations
offer logical information in quantitative form which we may use as we deem appropriate to improve our actual posterior
judgements.
5
3.3
Adjusted variance
We define the adjusted version of X given D, [X/D], to be the ‘residual’ form
[X/D] = X − E D (X ).
Adjusted quantities obey the following properties:
1.
E([X/D]) = 0;
(6)
Cov([X/D], E D (X )) = 0.
(7)
2.
We write X as the sum of the two uncorrelated components
X = [X/D] + E D (X ),
so that we can split Var(X ) as
Var(X ) = Var([X/D]) + Var(E D (X )).
The variance of the adjusted version of X , or the adjusted variance, Var D (X ), is defined to be
Var D (X ) = Var([X/D]) = E((X − E D (X ))2 ).
The value of Var D (X ) is determined by our prior variances and covariances as
Var D (X ) = Var(X ) − Cov(X, D)[Var(D)]−1 Cov(D, X )
(8)
The variance of X resolved by D, RVar D (X ), is defined as
RVar D (X ) = Var(E D (X )) = Cov(X, D)[Var(D)]−1 Cov(D, X ).
We therefore write the variance partition for X as
Var(X ) = Var D (X ) + RVar D (X )
(9)
In line with our various interpretations of belief adjustment, we may give corresponding interpretations to adjusted
variance. We may view Var D (X ) as:
• the ‘mean squared error’ of the estimator E D (X );
• a primitive expression, interpreted as we would a prior variance, but applied to the ‘residual variation’ when we have
extracted the variation in X ‘accounted for’ by D;
• a simple, easily computable upper bound on full Bayes preposterior risk, under quadratic loss, for any full prior
specification consistent with the given mean and variance specifications;
• within the more general view of the foundations, the adjustment variance attaches directly to our posterior beliefs.
We quantify the effect of an adjustment by evaluating the resolution, R D (X ), defined as
R D (X ) =
RVar D (X )
Var D (X )
=1−
.
Var(X )
Var(X )
(10)
If R D (X ) is near zero then either the collection D is not expected to be informative for X , relative to our prior knowledge
about X , or our beliefs have not been specified in sufficient detail to exploit the information contained in D.
Finally, we define the adjusted covariance, Cov D (X, Y ) to be
Cov D (X, Y ) = Cov([X/D], [Y/D]) = E((X − E D (X ))(Y − E D (Y ))).
6
3.4
Adjusting a collection of quantities
We have suggested how we might adjust our prior expectation for any one element of a collection B = {B1 , . . . , Br } using
observations on a collection D = {D1 , . . . , Dk }. When we evaluate a collection of adjusted expectations {E D (B1 ), …,
E D (Bk )}, we also implicitly evaluate the adjusted value for each element of hBi, the collection of linear combinations of
the elements of B, as, by the linearity of adjusted expectation (equation 2),
r
r
X
X
ED (
h i Bi ) =
h i E D (Bi ).
i=1
(11)
i=1
We now analyse changes in beliefs over hBi. We consider B, D as vectors, of dimension r and k, respectively. We
define the adjusted version of the collection B given D, [B/D], to be the ‘residual’ vector
[B/D] = B − E D (B).
The properties of the adjusted vector are as for a single quantity, namely
1.
E([B/D]) = 0,
(12)
Cov(E D (B), [B/D]) = 0,
(13)
the r dimensional null vector,
2.
the r × k null matrix.
Therefore, just as for a single quantity X , we partition the vector B as the sum of two uncorrelated vectors, namely
B = E D (B) + [B/D],
(14)
so that we may partition the variance matrix of B into two variance components
Var(B) = Var(E D (B)) + Var([B/D])
(15)
We call
RVar D (B) = Var(E D (B)),
the resolved variance matrix, for B by D. We call
Var D (B) = Var([B/D])
the adjusted variance matrix, for B by D.
E D (B), Var D (B) are calculated as in equations 1, 8, namely
E D (B) = E(B) + Cov(B, D)[Var(D)]−1 (D − E(D)),
Var D (B) = Var(B) − Cov(B, D)[Var(D)]
RVar D (B) = Cov(B, D)[Var(D)]
−1
3.5
−1
Cov(D, B),
Cov(D, B).
(16)
(17)
(18)
Adjusted belief structures
If we adjust each member of the base {B} by D, then we obtain a new base {[B1 /D], . . . , [Bk /D]}, the base of adjusted
versions of the elements of B. We call this the base {B} adjusted by D, written {B/D}. The belief structure with this base
is termed the adjusted belief structure of B by D and is written [B/D]. To simplify our notation, we also use [B/D] to
represent the vector ([B1 /D], . . . , [Bk /D]), where appropriate.
We may view [B/D] as representing a belief structure over the linear space h{B/D}i. However, it is also useful to
view [B/D] as an inner product space constructed over the linear space hBi but with the covariance inner product replaced
by the adjusted covariance inner product
(X, Y ) D = Cov D (X, Y ) = Cov([X/D], [Y/D]).
We now analyse the differences between the variance and the adjusted variance inner products.
7
3.6
Canonical directions
To assess how much information about [B] we expect to receive by observing D, we may first identify the particular linear
combination Y1 ∈ hBi for which we expect the adjustment by D to be most informative in the sense that Y1 maximises the
resolution R D (Y ) over all elements Y ∈ hBi with non-zero prior variance. (Note from equation 10, that maximising the
resolution is equivalent to minimising the ratio of adjusted to prior variance.) We may then proceed to identify directions for
which we expect progressively less information. This is equivalent to defining collections of canonical variables between
[B] and [D]. We make the following definition.
DEFINITION The j th canonical direction for the adjustment of B by D is the linear combination Y j which
maximises R D (Y ) over all elements Y ∈ hBi with non-zero prior variance which are uncorrelated a priori with Y1 , . . . , Y j−1 .
We scale each Y j to have prior expectation zero and prior variance one. The values
ri = R D (Yi ) = RVar D (Yi )
are termed the canonical resolutions. The number of canonical directions that we may define is equal to the rank,
r (B), of the variance matrix of the elements of B.
The quantities {Y1 , . . . , Yr (B) } are mutually incorrelated. It is also the case that the adjusted expectations, {E D (Y1 ), . . . , E D (Yr (B) )}
are also mutually uncorrelated, and each Yi is uncorrelated with each E D (Y j ), j 6= i.
The canonical resolutions may be calculated as the eigenvalues of the resolution matrix, TD , defined as
TD = [Var(B)]−1 RVar D (B).
(19)
We may calculate Y1 , . . . , Yr (B) by finding the normed eigenvectors, v1 , . . . , vr (B) , ordered by eigenvalues 1 ≥ r1 ≥
r2 ≥ . . . ≥ rr (B) ≥ 0, of TD , so that
Yi = viT B, Var D (Yi ) = 1 − ri .
The collection {Y1 , Y2 , . . .} forms a “grid” of directions over hBi, summarising the effects of the adjustment. We expect
to learn most about those linear combinations of the elements of B which have large correlations with those canonical
directions with large resolutions. The exact relation is as follows.
For any X in hBi,
R D (X ) =
rX
(B)
ci (X )ri ,
(20)
i=1
where
3.7
[Corr (X, Yi )]2
.
ci (X ) = Pr (B)
2
[Corr
(X,
Y
)]
j
j=1
The system resolution uncertainty
By analogy with the resolution for a single random quantity, we define the resolved uncertainty for the belief structure
[B] to be
rX
(B)
RU D (B) =
ri .
i=1
The total resolution is the sum of the resolutions for any collection of r (B) elements of hBi with prior variance one,
which are a priori uncorrelated. Note that if D is the empty set ∅, then
RU∅ (B) = r (B).
Where appropriate, we may therefore view r (B) as the prior uncertainty in the collection B, written as
U(B) = r (B),
namely the total uncertainty associated with any maximal collection of uncorrelated elements of hBi standardised to
unit variance. We define the system resolution for B to be the ratio of resolved to prior uncertainty for the collection,
namely
8
R D (B) =
RU D (B)
.
U(B)
The system resolution provides qualitatively similar information for the structure [B] to that expressed by the resolution
for a single quantity, X . R D (B) can be viewed as the “average” of the resolutions for each canonical direction, so that a
value near one implies that we expect substantial information about most elements of hBi, while a value near zero indicates
that there are a variety of elements for which the adjustment is not expected to be informative.
4
The observed adjustment
In the previous section, we constructed adjusted expectations given collections of observations. After we make the
observations and evaluate these adjustments, we express our overall changes in belief in ways which help us both to
identify qualitatively the most important changes between our prior and adjusted beliefs, and also to judge diagnostically
whether we should re-examine any aspects of our formulation. We proceed as follows.
4.1
Standardised observations
Each prior statement that we make describes our beliefs about some random quantity. When we actually observe this
quantity, we may compare what we expect to happen with what actually happens. A simple comparison is as follows.
For a single random quantity X , suppose that we specify E(X ) and Var(X ) and then observe value x. Using only our
limited belief specification, we evaluate the standardised observation defined as
x − E(X )
.
S(x) = √
Var(X )
S(x) has prior expectation zero and prior variance one. Thus, a very large absolute value for S(x) might suggest that
we have misspecified E(X ) or underestimated the variability of X , or misrecorded the value x, while a value near zero
might suggest that we have overestimated the variability of X . How large or small S(x) must be to merit attention depends
entirely upon the context, relating in large part to our confidence in our prior formulation.
4.2
Standardised adjustments
Suppose that we specify beliefs about a quantity, X , then adjust these beliefs by observation on a collection of quantities,
D. When we observe the actual data values,
D = d = (d1 , . . . , dk ),
then we may evaluate the random quantity E D (X ). The value which is obtained is denoted by Ed (X ). We apply the
standardisation operation to Ed (X ), defining the standardised adjustment as
Ed (X ) − E(X )
Sd (X ) = S(Ed (X )) = √
.
RVar D (X )
The value of Sd (X ) may suggest that our beliefs about X appear to be more or less affected by the data than we had
expected. Very large changes may raise the possibility that we have been overly confident in describing our uncertainty,
very small changes that we have been overly modest in valuing our prior knowledge about the value of X .
Such diagnostics provide us with qualitative and quantitative information. If our observations suggest to us substantially
new beliefs, then presumably it will be of interest to us to know this. (For example, we may appear to have made a great
discovery simply because of a blunder in our programming). Even when no simple explanation of a possible discrepancy
occurs to us, it will usually be of interest to identify which aspects of our beliefs have changed by substantially less or
more than we had expected. Such diagnostics are of particular importance when we make very large collections of belief
adjustments, so that we need simple, automatic methods to call our attention to particular assessments which we might
usefully re-examine.
9
4.3
Canonical standardised adjustments
When we adjust a collection, B, of random quantities by a further collection D, there are many standardised adjustments
that we may evaluate. A systematic collection of such consistency checks on our specification is provided by evaluating
the standardised value for each of the canonical directions, Yi , for the adjustment. We term these values the canonical
standardised adjustments defined as
Sd (Yi ) =
Ed (Yi ) − E(Yi )
.
√
ri
(21)
There are two types of diagnostic information given by these values. Quantitatively, any aberrant value may require
scrutiny. Qualitatively, we may look for systematic patterns. For example, as we expect larger changes in belief for the
first canonical directions than for subsequent directions, a particularly revealing pattern would be a sequence of decreasing
absolute values, which might suggest qualitatively a false prior classification between the more and the less informative
directions.
4.4
The bearing of the adjustment
Each evaluation that we have so far discussed assesses the change in belief for a single element of hBi. We now summarise
our overall changes in belief over hBi, relative to our prior uncertainty. We make the following definition.
Definition The size of the adjustment of B given D = d is
(Ed (X ) − E(X ))2
.
X ∈hBi
Var(X )
Sized (B) = max
Note: There are various alternative scalings for the changes in belief which we can choose, each of which may be
analysed in a similar fashion to our suggested choice and provide useful insights into the belief revision. Our particular
choice leads to the construction of various quantities whose properties unify many of the interpretive and diagnostic features
of the belief revision, and is particularly helpful when we come to consider the adjustment of beliefs in stages.
We now identify the element, Zd (B), of hBi with the largest such change in expectation. Zd (B) is termed the bearing
for the adjustment, and is constructed as follows.
Definition The bearing for the adjustment of the belief structure [B] by observation of D = d is the element Zd (B)
in hBi defined by
Zd (B) =
rX
(B)
Ed (Ui )Ui ,
(22)
i=1
where U1 , . . . , Ur (B) are any collection of elements of hBi which are a priori uncorrelated, with variance one. (The
canonical components of Var(B) form one such collection and the canonical directions for the adjustment form another
when suitably scaled. Zd (B) does not depend on the choice of U1 , . . . , Ur (B) .)
The bearing is so named as it expresses both the direction and the magnitude of the change between prior and adjusted
beliefs, as follows:
1. for any X which is a priori uncorrelated with Zd (B), Ed (X ) = E(X );
2. if Md = αZd (B), then a bearing of Md would represent α times the change in expectation as would a bearing of
Zd (B), for every element of hBi.
3. these properties follow as
Ed (X ) − E(X ) = Cov(X, Zd (B)), ∀X ∈ hBi.
(23)
We may therefore deduce that Zd (B) is indeed the direction of maximum change in belief, and that
Sized (B) = Var(Zd (B)) =
rX
(B)
i=1
10
E d2 (Ui ).
(24)
4.5
The expected size of an adjustment
A natural diagnostic for assessing the magnitude of an adjustment is to compare the largest standardised change in
expectation that we observe to our expectation for the magnitude of the largest change, evaluated prior to observing D.
This expectation is assessed as follows.
E(Size D (B)) = E(Var(Z D (B))) = RU D (B),
(25)
(Z D (B) is the random element of hBi which takes the value Zd (B) if D takes value d.) Thus, the expected size of the
adjustment is equal to the resolved uncertainty for the structure. To compare the observed and expected values, we define
the size ratio for the adjustment of B by D to be
Sized (B)
Var(Zd (B))
=
.
(26)
E(Size D (B))
RU D (B)
We anticipate that the ratio will be near one. Large values of the size ratio suggest that we have formed new beliefs
which are surprisingly discordant with our prior judgements. Values near zero might suggest that we have exaggerated our
prior uncertainty.
The size ratio is essentially a ratio of variances. To determine some ‘critical size’ for this quantity, we would, at the
least, need to assess the variance of our variance statements, i.e. to make fourth moment specifications. For the present,
we treat the ratio as a simple warning flag drawing our attention to possible conflicts between prior and adjusted beliefs.4
Sr d (B) =
4.6
Data size
Earlier in this section, we considered standardised changes in various individual quantities. We then considered measures
of maximal discrepancy in adjusted expectation. We now combine these two assessments.
For any data vector D, we may construct the collection of linear combinations hDi. For any element F ∈ hDi, with
observed value f , we must have Ed (F) = f . Therefore the element of hDi with the largest standardised observation
f − E(F) 2
) ,
max ( √
F∈hDi
Var(F)
is precisely the bearing Zd (D) of the adjustment of D by D. We therefore define the size of the data observation D as
follows.
Definition The size of the data vector D = d is
Sized (D) = max (S( f ))2 .
F∈hDi
Sized (D) is as defined in subsection 4.4. As in that subsection, we may construct the quantity Zd (D) as
Zd (D) =
rX
(D)
u i Ui ,
(27)
i=1
where U1 , . . . , Ur (D) are any uncorrelated collection of elements of hDi, with prior variance one, and u i is the observed
value of Ui . Zd (D) has the property that for any F ∈ hDi
f − E(F) = Cov(F, Zd (D)).
We have
Sized (D) = Var(Zd (D)) =
rX
(D)
(28)
u i2 ,
i=1
4 As an example of the type of simple rule of thumb that might sometimes be of use, observe that were all the elements of D to be normally distributed,
then it would follow that
Pr (B) 2
ri
Var(Sr D (B)) = 2 Pri=1
.
(B)
( i=1 ri )2
In certain circumstances, we might find it useful to approximate the distribution of Sr D (B), for example by a distribution of form cX , where c is a
Pr (B)
Pr (B)
constant and X has a χ 2 distribution, with ν degrees of freedom. Matching the mean and variance suggests a choice of ν = ( i=1 ri )2 /( i=1 ri2 )
degrees of freedom and c = 1/ν.
11
and
E(Size D (D)) = r (D),
the rank of Var(D). The size ratio for data vector D = d is therefore
Sized (D)
Sr d (D) =
=
E(Size D (D))
Pr (D)
u i2
.
r (D)
i=1
Again, we expect this value to be near one. Values which are very large or very close to zero suggest similar possible
misspecifications to those for a general adjustment.
5
Adjusting beliefs in stages
We have described the adjustment of beliefs about a collection of quantities by observation of a further collection. Often,
we will want to explore the ways in which different aspects of the data and the prior specification combine to give the final
adjustment. (For example, we might be combining information of various different types collected in different places by
different people at different times.) We now consider which aspects of the data are most crucial to the final adjustment,
in order to produce efficient sampling frames and experimental designs, a priori, and to investigate diagnostically whether
the various portions of the observed data have similar or contradictory effects on our beliefs.
5.1
Partial adjustment of beliefs
Suppose that we intend to adjust our beliefs about a collection B = {B1 , . . . , Br } by observation of two further collections
D = {D1 , . . . , Dk } and F = {F1 , . . . , F j } of quantities. We adjust B by the collection (D ∪ F) (i.e. the collection
{D1 , . . . , Dk , F1 , . . . , F j }) but separate the effects of the subsets of data. Therefore, we adjust B in stages, first by D,
then adding F. We may show that the additional adjustment of B by F, given that we have already adjusted by D, is the
same as the adjustment of B by [F/D], the belief structure [F] adjusted by [D]:
E(D∪F) (B) − E D (B) = E[F/D] (B),
(29)
(This relation follows as adjusting F by D removes the ‘common variability’ between F and D.) We call
E[F/D] (B),
the (partial) adjustment of B by F given D. Note the following properties of partial adjustment.
1.
E(E[F/D] (B)) = 0.
(30)
E D∪F (B) = E D (B) + E[F/D] (B).
(31)
[B/(D ∪ F)] = [(B/D)/(F/D)].
(32)
Cov(E[F/D] (B), E D (B)) = Cov(E[F/D] (B), [B/(F ∪ D)])
= Cov([B/(F ∪ D)], E D (B)) = 0.
(33)
2.
3.
4.
12
5.2
Partial variance
In section 3, we split the vector B into two uncorrelated components, as
B = E D (B) + (B − E D (B)).
We further decompose (B − E D (B)), and write
B
= E D (B) + (E D∪F (B) − E D (B)) + (B − E D∪F (B))
= E D (B) + E[F/D] (B) + [B/F ∪ D].
(34)
The three components on the right hand side of the above equation are mutually uncorrelated. We may partition
Var D (B), the ‘unresolved variation’ from the adjustment by D, as
Var D (B) = Var(E[F/D] (B)) + Var (D∪F) (B).
(35)
The second term is the adjusted variance matrix of B by D ∪ F, and the first is the (partial) resolved variance matrix of
B by F given D, namely
RVar [F/D] (B) = Var(E[F/D] (B)).
Resolved variances are additive in the sense that
RVar (F∪D) (B) = RVar D (B) + RVar [F/D] (B).
For any X ∈ hBi, we assess the further reduction in ‘residual variation’ from adding F, given D, as the (partial)
resolution, namely
R[F/D] (X ) =
5.3
RVar [F/D] (X )
.
Var(X )
(36)
Partial canonical directions
We summarise the effects of the partial adjustment in a similar fashion to that for a full adjustment. We make the following
definition.
DEFINITION The j th partial canonical direction for the adjustment of B by F given D is the linear combination
W j which maximises R[F/D] (B) over all elements in hBi with non-zero prior variance which are uncorrelated with each
Wi , i < j, scaled so that each Var(W j ) = 1. The values
f i = R[F/D] (Wi )
are termed the partial canonical resolutions.
The partial canonical directions for F given D are evaluated exactly as are the canonical directions for D, as described
in subsection 3.6, but the eigenstructure is extracted from the partial resolution matrix
T[F/D] = [Var(B)]−1 RVar [F/D] (B).
The collection W1 , W2 . . . forms a “grid” of directions over hBi, summarising the additional effects of the adjustment.
Having adjusted by D, we expect to learn most additionally from F for those linear combinations of the elements of B
which have large correlations with those partial canonical directions with large resolutions. The exact relation is as before,
namely for any X ∈ hBi,
R[F/D] (X ) =
rX
(B)
ci (X ) f i ,
i=1
where
(Corr (X, Wi ))2
.
ci (X ) = Pr (B)
2
j=1 (Corr (X, W j ))
13
(37)
The system partial resolution is
R[F/D] (B) =
Pr (B)
i=1
fi
r (B)
The resolution is additive, namely
R D (B) + R[F/D] (B) = R D∪F (B).
When we have made the adjustment, in addition to evaluating canonical standardised adjustments for the adjustment
by D and by D ∪ F, we may obtain similar qualitative insights into the changes in adjustment by evaluating the partial
canonical standardised adjustments which are as in subsection 4.3 but applied to the adjustment by [F/D].
5.4
Representing the observed partial adjustment
When we observe the values of D and F, and so of [F/D], taking values
D = d, F = f, [F/D] = [ f /d],
then we may evaluate the size of the partial adjustment defined to be
(Ed∪ f (X ) − Ed (X ))2
(E[ f /d] (X ))2
= max
.
X ∈hBi
X ∈hBi
Var(X )
Var(X )
Size[ f /d] (B) = max
We create the bearing for the partial adjustment, as
Z[ f /d] (B) =
rX
(B)
E[ f /d] (Ui )Ui ,
i=1
for any collection U1 , . . . , Ur (B) mutually uncorrelated with unit prior variance. Z[ f /d] (B) satisfies the relation
Ed∪ f (X ) − Ed (X ) = E[ f /d] (X ) = Cov(X, Z[ f /d] (B)), ∀X ∈ hBi.
(38)
Therefore
Size[ f /d] (B) = Var(Z[ f /d] (B)),
which may be compared to the expected value, namely
E(Size[F/D] (B)) = RU[F/D] (B).
Replacing B by F in the above expressions allows us to define the corresponding partial data size of F given D, namely
the largest change
( f − E D (F))2
= Var(Z[ f /d] (F)).
F∈hFi
Var(F)
Size[ f /d] (D) = max
5.5
Path correlation
When we adjust beliefs in stages, the expected sizes of the respective adjustments are additive in the sense that
E(Size D∪F (B)) = E(Size D (B)) + E(Size[F/D] (B))
However, the observed sizes of the adjustments are not additive. We have
Zd∪ f (B) = Zd (B) + Z[ f /d] (B).
(39)
The size of each adjustment is the variance of the corresponding bearing. Therefore
Var(Zd∪ f (B)) = Var(Zd (B)) + Var(Z[ f /d] (B)) + 2Cov(Zd (B), Z[ f /d] (B))
so that
14
(40)
Sized∪ f (B) = {Sized (B) + Size[ f /d] (B)} + 2Cov(Zd (B), Z[ f /d] (B)).
Thus, while
E(Cov(Z D (B), Z[F/D] (B))) = 0,
the observed value of this covariance
Cov(Zd (B), Z[ f /d] (B))
may be taken to expresses the degree of support/conflict between the two collections of evidence in determining the
revision of beliefs. As a summary, we define the path correlation to be
C(d, [ f /d]) = Corr(Zd (B), Z[ f /d] (B))
If this correlation is near 1 then the two collections of data are complementary in that their combined effect in changing
our beliefs is greater than the sum of the individual effect of each collection. If the path correlation is near -1 then the two
collections are giving contradictory messages which give smaller overall changes in belief, in combination, than we would
expect from the individual adjustments with D and [F/D].
5.6
Adjustment in several stages
Now suppose that we intend to adjust B sequentially by the collections of quantities G 1 , G 2 , . . . , G m . We define the
cumulative collection
G [i] =
i
[
G j.
j=1
and denote the cumulative adjustment
E[i] (B) = EG [i] (B).
We may ‘partial out’ any stage of the adjustment, defining for any i > j the partial adjustment of B by G [i] given
G [ j] as
E[i/j] (B) = E[i] (B) − E[ j] (B) = E[Si
k= j+1
Gk /
Sj
k=1
Gk ]
(B)
Corresponding to the adjustment E[i] (B) is the bearing Z[i] (B). The bearing for the partial adjustment E[i/j] (B) is
Z[i/j] (B) = Z[i] (B) − Z[ j] (B).
As before, we have
E[i] (X ) − E[ j] (X ) = Cov(X, Z[i/j] (B)).
The bearing for the partial adjustment expresses the change in both magnitude and direction in beliefs between stages
[ j] and [i]. The i th stepwise partial adjustment, E[i/] (B) is
E[i/] (B) = E[i/i−1] (B) = E[G i /G [i−1] ] (B),
with bearing Z[i/] (B) = Z[i] (B)−Z[i−1] (B). We refer to the full sequence of stepwise adjusted bearings Z[1] (B), Z[2/] (B), . . . , Z[m/] (B)
as the data trajectory. For each j we may write
Z[ j] (B) = Z[1] (B) + Z[2/] (B) + . . . + Z[ j/] (B)
We have therefore have that
Size[ j] (B) = Size[1] (B) + Size[2/] (B) + . . . + Size[m/] (B) + 2(C[2] + . . . + C[ j] )
where
C[r ] = Cov(Z[r −1] (B), Z[r/] (B))
So to examine the ways in which the individual terms combine to determine the revision we must consider
15
• the prior expectation for each change to assess which subcollections of data are expected to be informative;
• the individual adjusted bearings Z[i/] (B) to identify the stages at which larger than expected changes in belief occur;
• the path correlations C[i] to see whether the evidence is internally supportive or contradictory.
6
The geometry of belief adjustment
Just as traditional Bayes methods derive their formal properties from the structure of probability spaces, Bayes linear
methods derive their formal properties from the linear structure of inner product spaces. We now describe this underlying
geometry.
6.1
Belief adjustment
We have defined a (partial) belief structure as follows:
We have a collection C = {X 1 , X 2 , . . .}, finite or infinite, of random quantities, each with finite prior variance. We
construct the vector space hCi consisting of all finite linear combinations
c0 X 0 + c1 X i1 + . . . + ck X ik
of the elements of C, where X 0 is the unit constant. Covariance defines an inner product (·, ·) and norm, over the
closure of the equivalence classes of random quantities which differ by a constant in hCi, defined, for X, Y ∈ hCi to be
(X, Y ) = Cov(X, Y ), kX k2 = Var(X ).
The space hCi with covariance inner product is denoted as [C], the (partial) belief structure with base {C}.
Belief adjustment is represented within this structure as follows:
We have a collection {C} = {B1 , B2 , . . . , D1 , D2 , . . .}, the base for our analysis. We construct [C] as above. We
construct the two subspaces [B] and [D] corresponding to bases {B} = {B1 , B2 , . . .} and {D} = {D1 , D2 , . . .}. We define
P D to be the orthogonal projection from [B] to [D]. Thus, for any X ∈ hBi, P D (X ) is the element of [D] which is closest
to X in the variance norm. This orthogonal projection is therefore equivalent to the adjusted expectation, i.e.
E D (X ) = P D (X ).
(41)
Thus the adjusted version of X is
[X/D] = X − P D (X ),
namely the perpendicular vector from X to the subspace [D]. The adjustment variance Var D (X ) is therefore equal to the
squared perpendicular distance from X to [D]. Further, as
X = [X/D] + PD (X )
and [X/D] is perpendicular to P D (X ), we have
kX k2 = k[X/D]k2 + kPD (X )k2
which is the variance partition expressed in equation 9.
If we adjust each member of {B} by D, we obtain a new base {[B1 /D], . . . , [Bk /D]}, which we write as {B/D}. We
use [B/D] to represent both the vector of elements of {B/D} and the adjusted belief structure of B by D.
Alternately, it is often useful to identify [B/D] as a subspace of the overall inner product space [B ∪ D], namely the
orthogonal complement of [D] in [B ∪ D].
Note from this latter representation that for any bases D and F we may write a direct sum decomposition of [D ∪ F]
into orthogonal subspaces as
[D ∪ F] = [D] ⊕ [F/D],
(42)
P[D∪F] (X ) = P[D] (X ) + P[F/D] (X ), ∀X ∈ hBi,
(43)
Therefore, we may write
where the two projections on the right hand side of equation 43 are mutually orthogonal. The variance partition for a
partial belief adjustment follows directly from this representation.
16
6.2
A comment on the choice of inner product
While it is natural to view the variance inner product as describing our uncertainties, we may choose any inner product
over hCi which describes relevant aspects of our beliefs as the starting point for our analysis. One particular choice that
is frequently useful is the product inner product,
(X, Y ) = E(X Y ).
This inner product does not set the unit constant X 0 to zero. We can represent our original expectations by means of
orthogonal projections onto the subspace generated by the unit constant as
E(X ) = P X 0 (X ).
(44)
Within this belief structure, the covariance inner product is simply the adjustment by the unit constant, so that the inner
product space that we have termed [B] above, under this representation is more fully expressed as [B/ X 0 ]. Equivalently
[B] is the orthogonal complement of X 0 in hBi under the product inner product. Usually, we suppress the prior adjustment
by X 0 for notational simplicity, illustrating our freedom to choose whichever inner product is appropriate to emphasise the
important features of a particular analysis.
6.3
Belief transforms
Geometrically, the effect of the belief adjustment may be represented by the eigenstructure of a certain linear operator TD
defined on [B]. This operator TD is defined to be
TD = PB PD
(45)
where PB , PD are the orthogonal projections from [D] to [B], and from [B] to [D], respectively.
TD is a bounded self-adjoint operator, as PB , PD are adjoint transforms, namely
(X, PB (Y )) = (PD (X ), Y ), ∀X ∈ [B], Y ∈ [D],
(46)
because both sides of the above equation are equal to (X,Y).
The operator TD is termed the resolution transform for B induced by D, as it represents the variance resolved for
each X by D as
RVar D (X ) = Cov(X, TD (X )),
(47)
as
RVar D (X ) = Var(PD (X )) = (PD (X ), PD (X )) = (X, PB (PD (X ))).
We may also evaluate the transform
S D = I − TD ,
where I is the identity operator on [B]. We term S D the variance transform for B induced by D, as adjusted covariance
is represented by the relation, for each X and Y in hBi, that
Cov D (X, Y ) = Cov(X, S D Y ),
(48)
or equivalently, in terms of the inner products over [B], as
(X, Y ) D = (X, S D Y ).
(49)
TD , S D are self-adjoint operators, of norm at most one. They have common eigenvectors, Yi , with eigenvalues
1 ≥ ri , si ≥ 0, where ri + si = 1.
From equation 47, we may deduce that, provided TD has a discrete spectrum, each canonical direction, Yi , of the
adjustment of B by D, is an eigenvector of TD , with eigenvalue ri , and conversely each eigenvector of TD is a canonical
direction of the adjustment. Thus the eigenstructure of TD summarises the effects of the adjustment over the whole structure
[B]. In particular, the resolved uncertainty may be written as
RU D (B) = Trace(TD ).
17
(50)
6.4
Comparing inner products
The variance transform and the resolution transform are particular examples of the general class of belief transforms.
Suppose that we specify two inner products {·, ·}1 ,{·, ·}2 , over hBi, derived perhaps from alternative prior formulations or
alternative sampling frames. Provided that
{X, X }2
= M12 < ∞,
X ∈hBi {X, X }1
sup
(51)
then we may define a bounded, self-adjoint transform T on hBi, under inner product {·, ·}1 , with norm M12 , for which
{X, Y }2 = {X, T (Y )}1 , ∀X, Y ∈ hBi.
(52)
T is termed the belief transform for {·, ·}1 , associated with {·, ·}2 . For example, the variance transform S D is obtained
by selecting {·, ·}1 to be the inner product Cov(X, Y ), and {·, ·}2 to be the adjusted covariance inner product Cov D (X, Y ),
so that
Cov D (X, Y ) = Cov(X, S D (Y )), ∀X, Y ∈ hBi.
Just as the eigenstructure of the variance transform summarises the comparison between the prior and adjusted variance
specification, so does the eigenstructure of a general belief transform summarise the comparison between any two inner
products. The ratio {X, X }2 /{X, X }1 will be large/ near one / small according as whether X has large components
corresponding to eigenvectors with large/ near one / small eigenvalues.
Belief transforms provide a natural way to compare sequences of inner products, as they are multiplicative. Let Ti j be
the belief transform for {·, ·}i associated with {·, ·} j . Then we have
T13 = T12 T23 ,
(53)
(operator multiplication is by composition, namely T12 T23 (X ) = T12 (T23 (X ))), as
{X, T13 (Y )}1 = {X, Y }3 = {X, T23 (Y )}2 = {X, T12 (T23 (Y ))}1 .
This relation allows us to decompose a particular comparison into constituent stages. For example, if we wish to adjust
[B] by [D ∪ F], then we may decompose the overall variance transform S[D∪F] , into the product
S[D∪F] = S D S(D)F ,
(54)
where S(D)F is the variance transform S F applied to the adjusted space [B/D], so that
Cov D∪F (X, Y ) = Cov D (X, S(D)F (Y )).
(55)
Such multiplicative forms offer a natural sequential construction for a complicated belief transform. They also allow us
to apply the collection of interpretive and diagnostic tools that we have developed to each stage of a belief comparison or
adjustment.
6.5
The bearing
By the Riesz representation for linear functionals, f is a bounded linear functional on [B] if and only if there is a unique
element Z f ∈ [B], for which
f (X ) = (X, Z f ), ∀X ∈ hBi.
The difference between the prior expectation E(X ) and the observed adjusted expectation Ed (X ) defines a linear
functional
f d (X ) = Ed (X ) − E(X ),
on [B]. Therefore by the Riesz representation, if f d (X ) is bounded on [B]5 , then there is a unique element Z d ∈ [B],
corresponding to f d (X ), for which
5 for example, f will automatically be bounded if D has a finite number of elements
d
18
Ed (X ) − E(X ) = f d (X ) = (X, Z d ) = Cov(X, Z d ).
This element is precisely the bearing as created in section 4, and the properties of the bearing may be deduced directly
from this representation. Note that in the preceding sections we have also used the Riesz representation to create the
bearing for two other functionals, namely the difference functional, Ed∪F (X ) − Ed (X ), and also the functional which
replaces each X by its observed value.
References
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[4] M. Goldstein. Temporal coherence. In J.-M. Bernardo et al., editors, Bayesian Statistics 2, pages 231–248. North
Holland, 1985.
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North Holland, Amsterdam, 1986.
[6] M. Goldstein. Adjusting belief structures. J. R. Statist. Soc., B:50:133–154, 1988.
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19